Armodue harmony: Difference between revisions
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: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2010-09-10 03: | : This revision was by author [[User:hstraub|hstraub]] and made on <tt>2010-09-10 03:34:40 UTC</tt>.<br> | ||
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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Chapter 3: | Chapter 3: | ||
=Creating scales with Armodue: modal systems= | =Creating scales with Armodue: modal systems= | ||
==Modal systems based on tetrachords and pentachords== | |||
There are many systems how to generate various scales in Armodue. The most important ones, however, are based on the formation of tetrachords or pentachords and their subsequent unions. By analogy with the tetrachords (modes) of the interval system that was already developed in ancient Greek music, the tenth of Armodue (the tempered octave) is divided into two intervals of seven eka each (seven eka have a width close to five semitones or a perfect fourth of the tempered system): a lower interval of seven eka and an upper interval of also seven eka. | |||
The two intervals constituted that way will limit the tetrachords or pentachords that will be constituted inside of them and will be disjoint with a central interval of two eka between - to sum up to 16 eka or the entire scope of the tenth of Armodue. | |||
Here is the distribution pattern of the tenth 1-1: | |||
1, 1#, 2, 2#, 3, 3#, 4, 5 - 5# - 6, 6#, 7, 7#, 8, 8#, 9, 1 | |||
The first interval of 7 eka, the lower one, spans the scope of notes between 1 and 5, the second interval of 7 eka, the upper one, includes notes from 6 to 1. | |||
Between the two intervals there is a disjunction of 2 eka (note 5 to 6). Once the two limiting intervals are established as hinges or fixed structures, we proceed to "find notes" within the two intervals. In practice, the task is to partition the scalar distances between notes 1-5 and 6-1 in various ways. 7 eka can, for example, be partitioned into 3 + 3 +1 eka obtaining the lower tetrachord 1, # 2, 4, 5 and/or the upper tetrachord 6, # 7, 9, 1. | |||
On the other hand, seven eka can be arranged in a pentachord whose formula (interval structure) is 2 + 2 + 2 + 1 eka. In this latter case, we will have a lower pentachord 1, 2, 3, 4, 5 and/or an upper pentachord 6, 7, 8, 9, 1. The lower and the upper interval do not necessarily have to by divided in the same way; each lower tetrachord/pentachord can be combined with any tetrachord/pentachord type for the upper half, to create a big variety of heptatonic (type tetrachord/tetrachord), octatonic (type tetrachord/pentachord) or nonatonic (type pentachord/pentachord) scales. | |||
Here is the table of all possible tetrachord/pentachord formulas (excluding formulas that contain conscutive successions of 1 eka, for their excessive chromatic quality) across a total of seven eka: | |||
TETRACHORDS | |||
Tetrachordal interval partition: 1 + 2 + 4 eka | |||
1, 2, 4 | |||
1, 4, 2 | |||
2, 1, 4 | |||
2, 4, 1 | |||
4, 1, 2 | |||
4, 2, 1 | |||
Tetrachordal interval partition: 1+ 3 + 3 eka | |||
1, 3, 3 | |||
3, 1, 3 | |||
3, 3, 1 | |||
Tetrachordal interval partition: 2 + 2 + 3 eka | |||
2, 2, 3 | |||
2, 3, 2 | |||
3, 2, 2 | |||
Tetrachordal interval partition: 1 + 1 + 5 eka | |||
1, 5, 1 | |||
PENTACHORDS | |||
Pentachordal interval partition: 1 + 2 + 2 + 2 eka | |||
1, 2, 2, 2 | |||
2, 1, 2, 2 | |||
2, 2, 1, 2 | |||
2, 2, 2, 1 | |||
Pentachordal interval partition: 1 + 1 + 2 + 3 eka | |||
1, 2, 1, 3 | |||
1, 3, 1, 2 | |||
2, 1, 3, 1 | |||
3, 1, 2, 1 | |||
1, 2, 3, 1 | |||
1, 3, 2, 1 | |||
Overall, we have 13 types of tetrachords and 10 types of pentachords at our disposition to create scales; the number of different realizable scales is therefore 23 (23 = 13 + 10) squared, that is something like 529 different scales (in the context of scales that are organized into two limiting intervals of seven eka with 2 eka between). | |||
Each of these scales is transposable to any key (the possibilities are: 8464 scales - the product of 529 scale types x 16 possible tonics). | |||
If for example we combine the tetrachord with formula 2, 2, 3 with the pentachord formula with 2,1,3,1 get the eight-note (octotonic) scale formed by the following notes (starting with note 1): | |||
1, 2, 3, 5 - 6, 7, 7#, 9, 1 | |||
To build scales with full awareness, we must be familiar with the listed 23 types of tetrachords and pentachords and thoroughly study their sound and quality. Only when we have fully mastered the structures at the base of the scales - precisely the tetrachords and pentachords - we can proceed to their mutual combination in the formation of scales. | |||
==Modal systems based on hexachords== | |||
XXX | XXX | ||
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XXX</pre></div> | XXX</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Armodue armonia</title></head><body><!-- ws:start:WikiTextTocRule: | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Armodue armonia</title></head><body><!-- ws:start:WikiTextTocRule:42:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><div style="margin-left: 1em;"><a href="#Armodue: basic elements of harmony">Armodue: basic elements of harmony</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><div style="margin-left: 1em;"><a href="#Two theses supporting the system">Two theses supporting the system</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><div style="margin-left: 2em;"><a href="#Two theses supporting the system-The supremacy of the fifth and and the seventh harmonic in Armodue">The supremacy of the fifth and and the seventh harmonic in Armodue</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --><div style="margin-left: 2em;"><a href="#Two theses supporting the system-The triple mean of the double diagonal / side of the square">The triple mean of the double diagonal / side of the square</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><div style="margin-left: 1em;"><a href="#The interval table">The interval table</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><div style="margin-left: 2em;"><a href="#The interval table-Qualitative categories of intervals">Qualitative categories of intervals</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><div style="margin-left: 2em;"><a href="#The interval table-The intervals of 1 eka and 15 eka">The intervals of 1 eka and 15 eka</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><div style="margin-left: 2em;"><a href="#The interval table-The intervals of 2 eka and 14 eka">The intervals of 2 eka and 14 eka</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><div style="margin-left: 2em;"><a href="#The interval table-The intervals of 3 eka and 13 eka">The intervals of 3 eka and 13 eka</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><div style="margin-left: 2em;"><a href="#The interval table-The intervals of 4 eka and 12 eka">The intervals of 4 eka and 12 eka</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><div style="margin-left: 2em;"><a href="#The interval table-The intervals of 5 eka and 11 eka">The intervals of 5 eka and 11 eka</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><div style="margin-left: 2em;"><a href="#The interval table-The intervals of 6 eka and 10 eka">The intervals of 6 eka and 10 eka</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><div style="margin-left: 2em;"><a href="#The interval table-The intervals of 7 eka and 9 eka">The intervals of 7 eka and 9 eka</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><div style="margin-left: 2em;"><a href="#The interval table-The interval of 8 eka">The interval of 8 eka</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --><div style="margin-left: 2em;"><a href="#The interval table-Gradation of harmonic tensions">Gradation of harmonic tensions</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --><div style="margin-left: 2em;"><a href="#The interval table-Arrangement of tones in the chords">Arrangement of tones in the chords</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --><div style="margin-left: 1em;"><a href="#Creating scales with Armodue: modal systems">Creating scales with Armodue: modal systems</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --><div style="margin-left: 2em;"><a href="#Creating scales with Armodue: modal systems-Modal systems based on tetrachords and pentachords">Modal systems based on tetrachords and pentachords</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --><div style="margin-left: 2em;"><a href="#Creating scales with Armodue: modal systems-Modal systems based on hexachords">Modal systems based on hexachords</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --><div style="margin-left: 1em;"><a href="#x&quot;Geometric&quot; harmonic constructions with Armodue">&quot;Geometric&quot; harmonic constructions with Armodue</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --><div style="margin-left: 1em;"><a href="#x&quot;Elastic&quot; chords">&quot;Elastic&quot; chords</a></div> | ||
<!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --></div> | |||
<!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Armodue: basic elements of harmony"></a><!-- ws:end:WikiTextHeadingRule:0 --><strong>Armodue: basic elements of harmony</strong></h1> | |||
<br /> | <br /> | ||
This is a translation of an article by Luca Attanasio. Original page in italian: <a class="wiki_link_ext" href="http://www.armodue.com/armonia.htm" rel="nofollow">http://www.armodue.com/armonia.htm</a><br /> | This is a translation of an article by Luca Attanasio. Original page in italian: <a class="wiki_link_ext" href="http://www.armodue.com/armonia.htm" rel="nofollow">http://www.armodue.com/armonia.htm</a><br /> | ||
| Line 344: | Line 409: | ||
Chapter 3: <br /> | Chapter 3: <br /> | ||
<!-- ws:start:WikiTextHeadingRule:32:&lt;h1&gt; --><h1 id="toc16"><a name="Creating scales with Armodue: modal systems"></a><!-- ws:end:WikiTextHeadingRule:32 -->Creating scales with Armodue: modal systems</h1> | <!-- ws:start:WikiTextHeadingRule:32:&lt;h1&gt; --><h1 id="toc16"><a name="Creating scales with Armodue: modal systems"></a><!-- ws:end:WikiTextHeadingRule:32 -->Creating scales with Armodue: modal systems</h1> | ||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:34:&lt;h2&gt; --><h2 id="toc17"><a name="Creating scales with Armodue: modal systems-Modal systems based on tetrachords and pentachords"></a><!-- ws:end:WikiTextHeadingRule:34 -->Modal systems based on tetrachords and pentachords</h2> | |||
<br /> | |||
There are many systems how to generate various scales in Armodue. The most important ones, however, are based on the formation of tetrachords or pentachords and their subsequent unions. By analogy with the tetrachords (modes) of the interval system that was already developed in ancient Greek music, the tenth of Armodue (the tempered octave) is divided into two intervals of seven eka each (seven eka have a width close to five semitones or a perfect fourth of the tempered system): a lower interval of seven eka and an upper interval of also seven eka.<br /> | |||
<br /> | |||
The two intervals constituted that way will limit the tetrachords or pentachords that will be constituted inside of them and will be disjoint with a central interval of two eka between - to sum up to 16 eka or the entire scope of the tenth of Armodue.<br /> | |||
<br /> | |||
Here is the distribution pattern of the tenth 1-1:<br /> | |||
<br /> | |||
1, 1#, 2, 2#, 3, 3#, 4, 5 - 5# - 6, 6#, 7, 7#, 8, 8#, 9, 1<br /> | |||
<br /> | |||
The first interval of 7 eka, the lower one, spans the scope of notes between 1 and 5, the second interval of 7 eka, the upper one, includes notes from 6 to 1.<br /> | |||
<br /> | |||
Between the two intervals there is a disjunction of 2 eka (note 5 to 6). Once the two limiting intervals are established as hinges or fixed structures, we proceed to &quot;find notes&quot; within the two intervals. In practice, the task is to partition the scalar distances between notes 1-5 and 6-1 in various ways. 7 eka can, for example, be partitioned into 3 + 3 +1 eka obtaining the lower tetrachord 1, # 2, 4, 5 and/or the upper tetrachord 6, # 7, 9, 1.<br /> | |||
<br /> | |||
On the other hand, seven eka can be arranged in a pentachord whose formula (interval structure) is 2 + 2 + 2 + 1 eka. In this latter case, we will have a lower pentachord 1, 2, 3, 4, 5 and/or an upper pentachord 6, 7, 8, 9, 1. The lower and the upper interval do not necessarily have to by divided in the same way; each lower tetrachord/pentachord can be combined with any tetrachord/pentachord type for the upper half, to create a big variety of heptatonic (type tetrachord/tetrachord), octatonic (type tetrachord/pentachord) or nonatonic (type pentachord/pentachord) scales.<br /> | |||
<br /> | |||
Here is the table of all possible tetrachord/pentachord formulas (excluding formulas that contain conscutive successions of 1 eka, for their excessive chromatic quality) across a total of seven eka:<br /> | |||
<br /> | |||
TETRACHORDS<br /> | |||
Tetrachordal interval partition: 1 + 2 + 4 eka<br /> | |||
1, 2, 4<br /> | |||
1, 4, 2<br /> | |||
2, 1, 4<br /> | |||
2, 4, 1<br /> | |||
4, 1, 2<br /> | |||
4, 2, 1<br /> | |||
Tetrachordal interval partition: 1+ 3 + 3 eka<br /> | |||
1, 3, 3<br /> | |||
3, 1, 3<br /> | |||
3, 3, 1<br /> | |||
Tetrachordal interval partition: 2 + 2 + 3 eka<br /> | |||
2, 2, 3<br /> | |||
2, 3, 2<br /> | |||
3, 2, 2<br /> | |||
Tetrachordal interval partition: 1 + 1 + 5 eka<br /> | |||
1, 5, 1<br /> | |||
<br /> | |||
PENTACHORDS<br /> | |||
Pentachordal interval partition: 1 + 2 + 2 + 2 eka<br /> | |||
1, 2, 2, 2<br /> | |||
2, 1, 2, 2<br /> | |||
2, 2, 1, 2<br /> | |||
2, 2, 2, 1<br /> | |||
Pentachordal interval partition: 1 + 1 + 2 + 3 eka<br /> | |||
1, 2, 1, 3<br /> | |||
1, 3, 1, 2<br /> | |||
2, 1, 3, 1<br /> | |||
3, 1, 2, 1<br /> | |||
1, 2, 3, 1<br /> | |||
1, 3, 2, 1<br /> | |||
<br /> | |||
Overall, we have 13 types of tetrachords and 10 types of pentachords at our disposition to create scales; the number of different realizable scales is therefore 23 (23 = 13 + 10) squared, that is something like 529 different scales (in the context of scales that are organized into two limiting intervals of seven eka with 2 eka between).<br /> | |||
<br /> | |||
Each of these scales is transposable to any key (the possibilities are: 8464 scales - the product of 529 scale types x 16 possible tonics).<br /> | |||
<br /> | |||
If for example we combine the tetrachord with formula 2, 2, 3 with the pentachord formula with 2,1,3,1 get the eight-note (octotonic) scale formed by the following notes (starting with note 1):<br /> | |||
<br /> | |||
1, 2, 3, 5 - 6, 7, 7#, 9, 1<br /> | |||
<br /> | |||
To build scales with full awareness, we must be familiar with the listed 23 types of tetrachords and pentachords and thoroughly study their sound and quality. Only when we have fully mastered the structures at the base of the scales - precisely the tetrachords and pentachords - we can proceed to their mutual combination in the formation of scales.<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:36:&lt;h2&gt; --><h2 id="toc18"><a name="Creating scales with Armodue: modal systems-Modal systems based on hexachords"></a><!-- ws:end:WikiTextHeadingRule:36 -->Modal systems based on hexachords</h2> | |||
<br /> | <br /> | ||
XXX<br /> | XXX<br /> | ||
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<hr /> | <hr /> | ||
Chapter 4: <br /> | Chapter 4: <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:38:&lt;h1&gt; --><h1 id="toc19"><a name="x&quot;Geometric&quot; harmonic constructions with Armodue"></a><!-- ws:end:WikiTextHeadingRule:38 -->&quot;Geometric&quot; harmonic constructions with Armodue</h1> | ||
<br /> | <br /> | ||
XXX<br /> | XXX<br /> | ||
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<hr /> | <hr /> | ||
Chapter 5: <br /> | Chapter 5: <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:40:&lt;h1&gt; --><h1 id="toc20"><a name="x&quot;Elastic&quot; chords"></a><!-- ws:end:WikiTextHeadingRule:40 -->&quot;Elastic&quot; chords</h1> | ||
<br /> | <br /> | ||
XXX</body></html></pre></div> | XXX</body></html></pre></div> | ||